Two-Dimensional Billiards Are Turing Complete

Abstract

We show that two-dimensional billiard systems can simulate universal Turing machines. Billiards serve as idealized models of particle motion with elastic reflections and arise naturally as limits of smooth Hamiltonian systems under steep confining potentials. By invoking the undecidability of the halting problem, originally established by Turing, our results show that undecidable trajectories arise in physically natural billiard-type models, including models associated with hard-sphere gases and with collision-chain limits in celestial mechanics.

22 pages, 7 figures. Title changed and other expository changes. Final version to appear in the Proceedings of the National Academy of Sciences